Letter from John Collins to John Wallis
Sir
1 To your Apology about Writing in English I have this to adde that we have very few Latin booksellers that trade beyond Sea and such as doe make a more quick and profitable returne of their Stock than to adventure it in printing of Later Mathematiques as you well know by Experience.
2 That tis the Designe of the Royall Societie to promote and encourage the printing of Mathemaicks and other bookes of Art in our owne tounge.
Whereas you have given an Account of the learned paines of Englishmen namely of Harriot, your owne workes Mr Newton, you may yet adde one Stone more to the Monument of the fame of English Writers namely
To Commend the deceased Dr Barrow for setting such a studious painfull and learned Successor as Mr Isaac Newton, and to give an account of his Optick and Geometrick Lectures, which amongst most knowing Geometers have obtamed great esteeme though here they have mett with hard fate, the booksellers that first undertooke the same failing and the greatest part of the Impression which could not cost them lesse than 4 shillings for an Exemplar of both Tracts, afterwards sold for 1 shilling 6 pence, the doing hereof will not only please Mr Newton but likewise the Universitie of Cambridge and possibly revive the Sale of a booke that is slow and by consequence encourage Stationers in future undertakings Concerning Leibnitz, you see he fell into the method of Mr Newtons infinite Series which the said Mr Newton himselfe graunts, by a new method of transformation of Curves by which the extraction of the rootes of simple Powers in Species is avoyded, whether this was not learnt or may not be derived from Dr Barrows Geometrick Lectures is the question, also the said Leibnitz avoyds the Extraction of the rootes of adfected æquations in Species by an improoved method of Tangents, he is a most learned ingenious man, a Member of the Society, and now lately sent for home and admitted a Member of the Privy Councell of the Duke of Hannover: to explaine his doctrine more largely than himselfe could have leisure to doe, doubtlesse will not be unpleasing either to himselfe or to the R. Society who may order it to be printed with due respect to the author without incurring any censure for publishing what was occasioned by his private Letters, in the interim the same Compositor is employed to sett a Booke about Pallaviconos Religion of the Roman Court recommended by your Son and now licensed as also your English Exercises about Harriot &c which shall be sent you sheete by sheete, and if this retards, the other to proceed
<1v>Sir
I am loath to incurre your displeasure, but yet must take liberty to tell you some things concerning your intended Explanation of Mr Newtons Series
If had been so minded, I could about 9 yeares since namely at the beginning of 1669 have imparted to you a full treatise of his of that Argument but did not, in regard you lye under a censure from diverse for printing discourses that come to you in private Letters without permission or consent as is said of the parties concerned, Mr Newton last yeare sent up these Letters*, * you have seen with particular leave upon my importunity to print the same, and I seeing you therein mentioned imparted the first Letter to you (which if I had not I beleive you had not seen either to this day)
In your narrative you say Mr Newton began to fall into these methods in 1669 or 1670, whereas in the larger Letter he tells you he seemed delighted hisce ventis namely in Calculating Logarithmes and Van Ceulens Numbers in his retirement from the University in the Plague yeare in 1665, and in 1666 he writt the treatise above mentioned, all the account you can give out of those Letters is but very slender in relation to his performances, he intends a full treatise of Algebra consisting of these Parts according to the best of my {apps}
1 an Introductory part from Kinckhuysen out of low Dutch turned by Mercator into Latin, which he bought and is so excellent, that it comprehends many of Huddens reductions, and those mentioned by Dary at the end of his tract of Interest & some others to which Mr Newton added much of his owne
2 A discourse about bringing Problemes to an Æquation with a Collection of diverse notable ones
3 A Treatise about the Construction of Problemes and Æquations which I have seen, all Solid Problems viz those of 4 and 3 Dimensions are solved by ayd of one Constant Circle (if so desired) supposed to be intersected by Conick Sections, the description whereof is avoyded by helpe of mooveable angles that give the severall Points of Intersection sought, other Equations betweene the 5 and 9 degree he performes by ayd of a Cubicall Parabola that being once described in like manner remaines constant, and is to be intersected by a Conick Section the description whereof is avoyded as before &c
he hath also diverse tentative Constructions for Cubicks and Biquads from Plaine Geometry
4 A Discourse concerning the severall kinds of infinite Series considering which kinds are most convincing and fitt for Calculation, and which for Construction and Demonstration, of this Argument and of the whole buisinesse of Series he hath written a new and large treatise since that above mentioned, and hath performed aboundantly more than is either mentioned or can be guessed from the Letters above mentioned
5 A Treatise de Locis
6 The same applyed to Dioptriques concerning the worth of both which Dr Barrow affirmed he was not only surprized but others would thinke it incredible
Scotus owes debt to Aristotle, yet doth him plainer make
Therefore should he in Scots debt be, if there is no mistake
Mr Gregory having but one of Mr Newtons Series's sent him namely that for the Zone of a Circle after some study fell into the method, and began to be prurient about publishing something concerning it, as being offended I did not publish his Solution of Keplars Probleme, which I would not doe as knowing Mr Newton's Series's were made use of therein and yet he had a good right so to doe for he really advanced the Doctrine, namely after a few tearmes of a Series were attained he had approaches for attaining the Sum of a great many more, moreover he could give a fractionate part of some pure but high Power of the root of a Series equall to the Sum of as many tearmes therein as were desired 
These Series of Mr Newton did not hinder Mr Gregory from prosecuting his owne Converging Series, as more proper for Construction though worse for Calculation, who when he was last here that much after the Same manner as he had streightned an Arch of a Circle which I formerly accquainted you with he could streighten most other Curves, and particularly had streightened the Ellipticall Parabolic all and Hyperbolicall Lines which Dr Pell saith he can doe by a Canon of Sines & Tangents, the which indeed will remoove the use but not the necessity of Series from whence to derive such Canons and this being attained addes a topstone to this most excellent doctrine.
After Mercator had published his Logarithmotechnia you adde a Series better than his which you now claime, What sayes Gregory to this, namely that he emptyed both Mercators and your Series of each other tearme, and what Mercator, namely in an Appendix now at the Presse, he shewes by his owne methods rejecting these advantages how to make a whole Canon of Logarithmes by Addition, you may therefore omitt such claime, without any Dispendium of renowne
<2v>Sir In sum the tendency of what you have writt, is to show the world that Mr Newton's Series were derived from yours he graunts, sayth so himselfe he attained his Seriess by 3 Methods the first as I remember for I have not as yet a Coppy of the larger letter was by interpoling yours where your selfe gave over, and afterwards he forsooke this method having falne into two ones of his owne
To Dr Wallis
